Lattice Agreement in Message Passing Systems

Abstract: This paper studies the lattice agreement problem and the generalized lattice agreement problem in distributed message passing systems. In the lattice agreement problem, given input values from a lattice, processes have to non-trivially decide output values that lie on a chain. We consider the lattice agreement problem in both synchronous and asynchronous systems. For synchronous lattice agreement, we present two algorithms which run in $\log f$ and $\min {O(\log^2 h(L)), O(\log^2 f)}$ rounds, respectively, where $h(L)$ denotes the height of the {\em input sublattice} $L$, $f < n$ is the number of crash failures the system can tolerate, and $n$ is the number of processes in the system. These algorithms have significant better round complexity than previously known algorithms. The algorithm by Attiya et al. \cite{attiya1995atomic} takes $\log n$ synchronous rounds, and the algorithm by Mavronicolasa \cite{mavronicolasabound} takes $\min {O(h(L)), O(\sqrt{f})}$ rounds. For asynchronous lattice agreement, we propose an algorithm which has time complexity of $2 \cdot \min {h(L), f + 1}$ message delays which improves on the previously known time complexity of $O(n)$ message delays. The generalized lattice agreement problem defined by Faleiro et al in \cite{faleiro2012generalized} is a generalization of the lattice agreement problem where it is applied for the replicated state machine. We propose an algorithm which guarantees liveness when a majority of the processes are correct in asynchronous systems. Our algorithm requires $\min {O(h(L)), O(f)}$ units of time in the worst case which is better than $O(n)$ units of time required by the algorithm of Faleiro et al. \cite{faleiro2012generalized}.